Some Nonlinear Problems

Abstract

Although we have derived universal results that relate stresses to the applied loads and geometry for axially loaded rods and inflated thin-walled cylinders and spheres (see Sections 3.3–3.5 of Chapter 3), all other results for stresses and deformations have been restricted to linear, elastic, homogenous, and isotropic (LEHI) behaviors under small strains. Such results for LEHI behaviors are very important in bone mechanics, many applications involving biomaterials, and the design of experimental systems. Nonetheless, as noted in Chapter 2, the behavior of cells, soft biological tissues, and elastomers used in biomedical applications typically exhibit a nonlinear stress-strain behavior over large deformations and, consequently, there is a need for additional theoretical frameworks. In particular, LEHI behavior is characterized by two linearities: a linear relation between stress and strain [e.g., Eq. (2.69)] and a linear relation between strain and displacement gradients [e.g., Eq. (2.45)]. We need to avoid both linearizations in soft tissue mechanics. The interested reader is referred to Humphrey (2002) for a more complete treatment of the nonlinear theories, but here we provide a simple, brief introduction.